Approximating x*log(x) as x->0

The approximation is trivial if [itex]C < 1[/itex], because then [itex]x^{1-C}\log(x)[/itex] would approach zero, and [itex]A[/itex] and [itex]B[/itex] could be chosen to be almost anything (only [itex]C<B[/itex] needed). But if [itex]1\leq C[/itex], then the approximation could have some content. Obviously conditions

[tex]
A < 0 < B < 1
[/tex]

should hold, because [itex]x\log(x) < 0[/itex] when [itex]0<x<1[/itex], and [itex]D_x(x\log(x))\to \infty[/itex].

update:

I see these numbers do not exist, because if they did, then also [itex]\log(x)[/itex] could be approximated with some [itex]\alpha x^{\beta}[/itex] where [itex]\beta <0[/itex].